In Abelian l-adic Representations and Elliptic Curves (1968), J. P. Serre showed that the adelic representation $$\rho_{E}\colon G_K \to \mathrm{GL}(\hat{\mathbb{Z}}^2)$$ associated to an elliptic curve $E/K$ over a number field $K$ has open image. To do it, he uses Shafarevich's Theorem on the finiteness of isomorphism classes of elliptic curves in a given isogeny class to show that the $\ell$-adic representation $$\rho_{E,\ell}\colon G_K \to \mathrm{GL}(T_\ell(E))$$ is irreducible for all $\ell$ and that the mod $\ell$ representation $$\bar{\rho}_{E,\ell}\colon G_K \to \mathrm{GL}(E[\ell])$$ is irreducible for almost all $\ell$.
My question is, do we now have a method of proving this theorem without using Shafarevich's Theorem? The latter depends on Siegel's Theorem, which depends on Roth's Theorem in Diophantine Geometry.
Best Answer
First, you forgot to assume that $E$ does not have CM. However, this actually suggests a difficulty in a Shafarevich-free proof.
Let $K = \mathbf{Q}(\sqrt{-1})$, and let $C/K$ be an elliptic curve with CM by $\mathbf{Z}[\sqrt{-1}]$. Now consider the following thought experiment. Can you rule out the existence of an elliptic curve $E/K$ without CM such that $\rho_{E,\ell} = \rho_{C,\ell}$ for all primes $\ell$?
This is certainly implied by the Tate conjecture (in a case proved by Faltings), but not only is this harder than the original proof, it also really uses/implies a (generalization of) Shafarevich's conjecture. Certainly $E$ admits isogenies $E \rightarrow E'$ of degree $p$ for any prime $p$ which splits in $K$, but ruling this out is exactly Shafarevich again. I'm not sure you can overcome this obstacle.
On the other hand, it is elementary to (essentially) reduce to this case, basically using Serre's original argument. Namely, one reduces to the case that the $\rho_{E,\ell}$ are abelian, and a classification of crystalline characters of the right weight (plus purity) essentially reduces to this CM-like case.